Question
Who are the children of Vishnu and Lakshmi?
Answer
As far as I know, the only time that children of Lakshmi are mentioned in Hindu scriptures is in verses 11 - 12 of the Sri Suktam, a hymn from the Khilani (apocrypha or later portions) of the Rig Veda:
We are the progeny of our forefather, Sage Kardama, who is one of the sons of Goddess Lakshmi. We invoke that Sage Kardama to install in his family the Universal Mother, Mahalakshmi, who is decked with the garland of lotuses. So be it. We invoke another son of Lakshmi named Chikleeta. May he dwell in our home and may his mother, Mahalakshmi, dwell in our family.
There's almost no information known beyond that. The sage Kardama is usually thought of as a son of Brahma, born from Brahma's shadow, as mentioned in this verse of the Srimad Bhagavatam. The only other connection between Lakshmi and Kardama is a story about how when Lakshmi emerged out of the churning of the ocean, she took sage Kardama as her adopted father, not son, as a reward for his Tapasya. And I don't think Chiklita is mentioned in any other scriptures.
The meanings of the names Kardama and Chiklita both refer to moist or wet things, so that might also be related to Lakshmi's emergence from the churning of the ocean,
One more thing that you may want to know is that Kardama and Chilkita are apparently two of the seers of the verses in the Sri Suktam, according to this webpage:
If we consider the entire Sukta as a whole, then the Rishis are: Each mantra of this Sukta has a different Rishi (seer), Chandas (metre), Devi (deity) and Viniyoga (ritual application). Ananda, Chikleeta, Kardama, Shreeda and Indira, who are said to be the sons of goddess Lakshmi.
But I'm not sure whether the claim that these are all sons of Lakshmi has any scriptural basis.
EDIT: This excerpt from the Vayu Purana lists two named more sons of Vishnu and Lakshmi, "Bala (Strength) and Utsaha (Energy)". And it also says the drivers of vimanas were born to her as "mental sons", and that she had two daughters, Ayati and Niyati, who became daughters-in-law of Brighu (who is Lakshmi's adopted father!).
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